On the randomized complexity of monotone graph properties

  Gröger Hans Dietmar

 Let be the number of questions of the form 'Does the graph contain the edge ?' that have to be asked in the worst case by any randomized decision tree algorithm for computing an -vertex graph property . For non-trivial, monotone graph properties it is known, that the deterministic complexity is (see []). R. Karp [] conjectured, that this bound holds for randomized algorithms as well. As far as this conjecture we know the following results. The best uniform lower bound for all non-trivial, monotone graph properties is due to P. Hajnal [].

 No non-trivial, monotone graph property is known having a randomized complexity of less than . Some properties have been proven to have complexity of (see A. Yao []).

 In this paper we refine the idea of Yao. This leads to a further improvement in the reductions of arbitrary graph properties to bipartite graph properties. (see [], []) and yields a uniform lower bound for the subgraph isomorphism properties of . Furthermore we show, that a large variety of isomorphism properties as well as -colourability require questions.

 Gyenizse Pal 1996. Szeptember 4. Szerda 13:59:59 MET DST